Method and device for rapidly calculating sensitivity of linear periodic time-varying system based on matrix sparse technology

Abstract

The invention discloses a rapid calculation method and device for the sensitivity of a linear periodic time-varying system based on a matrix sparse technology, and belongs to the technical field of modeling analysis of the linear periodic time-varying system.The rapid calculation method comprises the steps that an LTP system is converted into an equivalent LTI system, and the eigenvalue, the left eigenvector and the right eigenvector of the LTP system are obtained; based on a functional derivative method, obtaining a sensitivity analytical expression of the LTP system characteristic value to the parameter; according to the influence mechanism of the parameters on the system matrix, obtaining an analysis form of the influence of the parameters on the LTP system matrix in the sensitivity analysis expression; and converting a partial derivative of a system matrix to a steady-state trajectory into sparse calculation of a vector field Hessian matrix by using a matrix sparse technology, thereby obtaining a calculation result of the characteristic value sensitivity of the LTP system. According to the method, the sparse characteristic of the system matrix in the PEPS is utilized, the calculated amount and the storage requirement of the system matrix on trajectory partial derivative solving are reduced, and therefore calculation of the LTP characteristic value sensitivity is accelerated.

Description

Technical Field

[0001] The present invention belongs to the technical field of linear periodic time-varying system modeling and analysis, and more specifically, relates to a method and device for quickly calculating the sensitivity of a linear periodic time-varying system based on matrix sparsity technology. Background Art

[0002] As electricity demand continues to grow, the primary energy source for power systems is gradually shifting from traditional fossil fuels to renewable energy. To support the application of renewable energy in all aspects of power generation, transmission, and consumption, the widespread integration of power electronics technology has become essential. Consequently, modern power systems are increasingly dominated by power electronics, forming power electronic-dominated power systems (PEPS).

[0003] However, small-signal stability issues characterized by broadband oscillations pose a serious threat to the safe operation of PEPS. General time-varying characteristics with multiple frequency components are becoming increasingly common, such as the dynamic coupling between positive and negative sequence control and the switching processes of submodules in modular multilevel converters. These time-varying characteristics lead to so-called nonlinear time-periodic (NLTP) systems, characterized by periodic trajectories in the steady state. While it is possible to linearize the NLTP system on the periodic trajectory to obtain a linear time-varying (LTP) system, and to perform small-signal stability analysis by calculating eigenvalues ​​based on detailed system topology and parameters, revealing the stability patterns remains a significant challenge. Therefore, it is crucial to deeply analyze how various parameters affect the oscillation modes in PEPS, namely, to conduct eigenvalue sensitivity analysis of the LTP system.

[0004] In recent years, eigenvalue sensitivity analysis for LTP systems has been further developed, and its analytical form has been derived. However, the complexity of calculating eigenvalue sensitivities for LTP systems has severely limited their practical applications. This is primarily due to the need to calculate the partial derivatives of the system matrix with respect to the trajectory, resulting in computational and storage requirements that grow cubically with system size, making them prohibitive for large-scale PEPS. Summary of the Invention

[0005] In response to the above-mentioned defects or improvement needs of the prior art, the present invention provides a method and device for quickly calculating the sensitivity of a linear periodic time-varying system based on matrix sparsity technology. The purpose is to reduce the computational complexity and storage requirements of the partial derivatives of the system matrix for the trajectory based on the sparse characteristics of the system matrix, thereby accelerating the calculation of the sensitivity, thereby solving the technical problem of the complex calculation of the eigenvalue sensitivity of the existing LTP system.

[0006] To achieve the above object, according to one aspect of the present invention, a method for fast calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology is provided, comprising:

[0007] Step S1, based on the Floquet-Lyapunov method, the LTP system is transformed into an equivalent LTI system through a periodic transformation matrix, and the stationary matrix of the LTI system is eigendecomposed to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system;

[0008] Step S2, obtaining an analytical expression of the sensitivity of the LTP system eigenvalue to the parameters based on the functional derivative method;

[0009] Step S3, distinguishing the direct and indirect effects of the parameters on the system matrix according to the mechanism of the effects of the parameters on the system matrix, and obtaining the analytical form of the LTP system matrix affected by the parameters in the sensitivity analytical expression;

[0010] In step S4, the matrix sparsity technique is used to convert the partial derivative of the system matrix with respect to the steady-state trajectory into a sparse calculation of the vector field Hessian matrix, thereby obtaining the calculation result of the eigenvalue sensitivity of the LTP system.

[0011] Preferably, the sparse calculation of converting the partial derivative of the system matrix with respect to the steady-state trajectory into the vector field Hessian matrix in step S4 includes:

[0012] Re-express the partial derivative of the system matrix with respect to the trajectory as And calculate the upper or lower triangular matrix of the Hessian matrix, where the Hessian matrix is the vector field f of the system i The second-order partial derivative with respect to the system's state vector y.

[0013] Preferably, in step S1, converting the LTP system into an equivalent LTI system by using a periodic transformation matrix includes:

[0014] Get LTP System Where x(t)=[x1(t),x2(t),...,x n (t)] T represents the column vector composed of n state variables of the system at time t, A(t) represents the periodic system matrix, and T represents the basic period of the system;

[0015] The periodic transformation matrix P(t) transforms the LTP system into an LTI system, and the systems before and after the transformation have the same stability: Among them, v(t) represents a set of column vectors consisting of new state variables, Q is a time-invariant n-order square matrix, and P(t) is an n-order square matrix with a period of T, thereby realizing the conversion of the time-invariant system matrix A(t) into a constant matrix Q.

[0016] Preferably, in step S1, the LTI system's constant matrix is ​​subjected to eigendecomposition to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system, including:

[0017] Perform eigendecomposition on the constant matrix Q: VQU=Λ=diag(λ1,λ2,...,λ n ), where V and U represent the left and right eigenvectors of the constant matrix Q, respectively, and the diagonal matrix Λ represents the Floquet characteristic index λ of the LTP system. i (i=1,2,...,n), i.e., LTP eigenvalue;

[0018] Combining the transformation matrix P(t) with the right eigenvector U of the Q matrix, we can construct the coordinate transformation R(t)=P(t)U. The transformation matrix R(t) realizes that the system matrix A(t) remains unchanged and is converted into the eigenvalue matrix Λ of the LTP system: Where L(t) = R -1 (t) and R(t) represent the left and right eigenvectors of the LTP system, respectively.

[0019] Preferably, the analytical expression for obtaining the sensitivity of the LTP system eigenvalue to the parameter in step S2 includes:

[0020] In the LTP system, the mode λ k The first-order variation of

[0021] δλ k =λ k [α(t)+δα(t)]-λ k [α(t)],

[0022] According to the functional derivative, the analytical expression of sensitivity is obtained as follows:

[0023]

[0024] When only time-invariant parameters are considered, In unit 1 form, the expression is

[0025]

[0026] Preferably, the analytical form of the LTP system matrix obtained in step S3 includes:

[0027] Denote the system matrix as A(y ss (α),α), where y ss is the steady-state solution;

[0028] The influence of parameter α on the system matrix is ​​divided into indirect influence and direct influence, where the indirect influence A IThe parameter affects the steady-state solution y ss (α) further affects the system matrix and directly affects A E The parameters directly affect the system matrix without affecting the steady-state solution;

[0029] Based on the parameters of A I With A E The influence of the two parts, the analytical form of the parameter influence LTP system matrix is ​​obtained as follows:

[0030] According to another aspect of the present invention, there is provided a device for quickly calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology, comprising:

[0031] The first processing module, based on the Floquet-Lyapunov method, transforms the LTP system into an equivalent LTI system through a periodic transformation matrix and performs eigendecomposition on the stationary matrix of the LTI system to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system;

[0032] The second processing module obtains the analytical expression of the sensitivity of the LTP system eigenvalue to the parameters based on the functional derivative method;

[0033] The third processing module distinguishes the direct and indirect effects of parameters on the system matrix according to the mechanism of their influence on the system matrix, and obtains the analytical form of the parameter-affected LTP system matrix in the sensitivity analytical expression;

[0034] The fourth processing module uses matrix sparsity technology to convert the partial derivatives of the system matrix with respect to the steady-state trajectory into a sparse calculation of the vector field Hessian matrix, thereby obtaining the calculation result of the eigenvalue sensitivity of the LTP system.

[0035] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0036] 1. The present invention provides a fast calculation method for the sensitivity of a linear periodic time-varying system based on matrix sparsity technology. It utilizes the sparse characteristics of the system matrix in PEPS to reduce the computational complexity and storage requirements of the partial derivative of the system matrix for the trajectory, thereby accelerating the calculation of the LTP eigenvalue sensitivity.

[0037] 2. The present invention provides a method for quickly calculating the sensitivity of a linear periodic time-varying system based on matrix sparsity technology. Through the Floquet-Lyapunov method, the LTP system is converted into an equivalent linear time-invariant (LTI) system, so that the eigenvalues ​​and eigenvectors of the LTP system can be directly calculated. This conversion simplifies the analysis of the LTP system and provides the necessary mathematical basis for subsequent sensitivity calculations.

[0038] 3. This invention provides a method for rapidly calculating the sensitivity of linear periodic time-varying systems based on matrix sparsity technology. Using functional derivatives, it establishes the relationship between the sensitivity of LTP system eigenvalues ​​and the system matrix, eigenvectors, and system parameter variations. This analytical expression enables direct calculation of the sensitivity of eigenvalues ​​to system parameters, providing a theoretical foundation for subsequent system analysis and optimization.

[0039] 4. This invention provides a fast sensitivity calculation method for linear periodic time-varying systems based on matrix sparsity technology. By distinguishing between the direct and indirect effects of parameters on the system matrix, it more accurately describes the impact of parameter changes on the LTP system. This distinction enables more precise calculation of the sensitivity of the system matrix to the parameters, thereby improving the accuracy of the overall sensitivity analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 This is a flow chart of a method for quickly calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology of the present invention. DETAILED DESCRIPTION

[0041] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0042] according to Figure 1 As shown, the present invention provides a method for calculating the eigenvector of a linear periodic time-varying system, comprising the following steps:

[0043] S1, based on the Floquet-Lyapunov method, obtain the eigenvalues, left and right eigenvectors of the LTP system. Furthermore, for step S1, the NLTP system with general periodic coefficients can be described as:

[0044]

[0045] Among them, y(t)=[y1(t),y2(t),...,y n (t)] T Represents the system state vector, f=[f1,f2,...,f n ] T represents the vector field of the system.

[0046] By linearizing the NLTP system on a periodic trajectory, the corresponding LTP system can be obtained:

[0047]

[0048] Where x(t)=[x1(t),x2(t),...,x n (t)] T Represents the column vector composed of n state variables of the system at time t, A(t) represents the periodic system matrix, and T represents the basic period of the system.

[0049] According to the Floquet-Lyapunov method, there exists a periodic transformation matrix P(t) that transforms the LTP system into the LTI system, and the systems before and after the transformation have the same stability:

[0050]

[0051] Among them, v(t) represents a set of column vectors consisting of new state variables, Q is a time-invariant n-order square matrix, and P(t) is an n-order square matrix with a period of T, which represents a special coordinate transformation that realizes the transformation of the system matrix A(t) from time-invariant to a constant matrix Q. The transformation matrix P(t) satisfies:

[0052]

[0053] Perform eigendecomposition on the constant matrix Q:

[0054] VQU=Λ=diag(λ1,λ2,...,λ n ) (5)

[0055] Where V and U represent the left and right eigenvectors of the constant matrix Q, respectively, and the diagonal matrix Λ represents the Floquet characteristic index λ of the LTP system. i (i=1,2,...,n), namely the LTP eigenvalue.

[0056] Furthermore, by combining the transformation matrix P(t) with the right eigenvector U of the Q matrix, we can construct the coordinate transformation R(t)=P(t)U. The transformation matrix R(t) realizes that the system matrix A(t) remains unchanged and is converted into the eigenvalue matrix Λ of the LTP system:

[0057]

[0058] Where L(t) = R -1 (t) and R(t) represent the left and right eigenvectors of the LTP system, respectively.

[0059] Arrange Equation (6) and express the eigenvalue matrix Λ as diag(λ1,λ2,...,λ n ), the eigenvector matrix is ​​expressed as R(t)=[r1(t),r2(t),...,r n(t)] and L(t)=[l1(t); l2(t);...;l n (t)], where r i (t) and l i (t) are the corresponding eigenvalues ​​λ i The right eigenvector (column vector) and left eigenvector (row vector) of , and then the corresponding LTP eigenvalue λ can be derived i The corresponding dynamic differential equation of the LTP eigenvector is:

[0060]

[0061] S2, based on the functional derivative method, the analytical expression of the sensitivity of the LTP system is derived. Furthermore, for step S2, in the LTP system, the mode λ k is a functional of the time-varying parameter α(t), so the first-order variation δλ of the mode can be k writing:

[0062] δλ k =λ k [α(t)+δα(t)]-λ k [α(t)] (9)

[0063] Here, δα represents a small change in the parameter.

[0064] According to formula (6), the system matrix with time-varying parameters is mapped to the time-invariant LTP eigenvalue, so the variation of the LTP eigenvalue δλ k is time-invariant. Further, δα is expressed as Indicates that, Indicates the form of change of the parameter under study, such as sinusoidal change sin(ωt), ε is the coefficient describing small changes. Then, the mode λ k Using the system matrix and eigenvector, Equation (9) can be rewritten as:

[0065]

[0066] In order to make the expression simpler, A(t) and r are omitted. k (t) and l k (t) Information about time t.

[0067] According to the properties of LTP eigenvector, Ar in formula (10) k and l k A satisfies the following relationship:

[0068]

[0069] Therefore, formula (10) can be rewritten as follows:

[0070]

[0071] The right side of the equal sign in Equation (12) can be divided into three parts: For the first term, the left and right eigenvectors l k and r k The product of is a constant value 1, independent of the parameter α(t), so the first term is equal to 0:

[0072] term1=λ k δ(l k ·r k )=0 (13)

[0073] For the second term, since the eigenvector r is generally k The second-order partial derivatives of α and t satisfy the continuity conditions, and the order of derivation can be exchanged, so we can get:

[0074]

[0075] Since the left eigenvector l k and the variation of the right eigenvector δr k is periodically time-varying, l k and δr k The time derivative of the product will not contain a DC component. Furthermore, due to δλ k is time-invariant, the time-varying components in the second and third terms cancel each other out. Therefore, Equation (12) can be reformulated as:

[0076] term1=λ k δ(l k ·r k )=0 (15)

[0077] For the second term, since the eigenvector r is generally k The second-order partial derivatives of α and t satisfy the continuity conditions, and the order of derivation can be exchanged, so we can get:

[0078]

[0079] Since the left eigenvector l k and the variation of the right eigenvector δr k is periodically time-varying, l k and δr k The time derivative of the product will not contain a DC component. Furthermore, due to δλ k is time-invariant, the time-varying components in the second and third terms cancel each other out. Therefore, Equation (12) can be reformulated as:

[0080]

[0081] The subscript 0 represents the DC component, indicating that only the DC component of the third term on the right side of the equation is retained.

[0082] According to the definition of functional derivative, Equation (17) can be rewritten as:

[0083]

[0084] Therefore, Equation (18) expresses the mode λ k The parameter α(t) of interest is in the form of When only time-invariant parameters are considered, is in the unit 1 form, formula (18) degenerates into:

[0085]

[0086] S3, based on the mechanism of the influence of parameters on the system matrix, the analytical form of the parameter-affected LTP system matrix is ​​derived. Further, in order to derive the analytical form of the parameter-affected LTP system matrix in equation (18): It is necessary to consider the mechanism by which the parameters affect the system matrix. According to formula (2), the steady-state trajectory y ss Substituting the Jacobi matrix of f with respect to y, we can obtain the form of the system matrix, which can be expressed as A(y ss (α),α). Therefore, the parameter α can be used to adjust the system matrix A(y ss (α),α) have an impact: one is an indirect impact, which mainly considers the parameters affecting the steady-state solution y ss (α) further affects the system matrix, such as operating parameters, which are implicit in A(y ss (α)), and use the symbol A I The other is direct influence, which mainly considers the case where the parameter does not affect the steady-state solution, such as the control parameter. In this case, the parameter is explicitly contained in A(α) and is represented by the symbol A E Indicates. Based on the parameters A I With A E The influence of the two parts, the functional derivative of the system matrix with respect to the parameters can be written as:

[0087]

[0088] S4, based on the matrix sparse technology, the analytical form of the partial derivative of the system matrix with respect to the trajectory is derived. The computational complexity is proportional to the cube of the system size n, which is unacceptable for large-scale PEPS. Therefore, the partial derivative of the system matrix with respect to the trajectory can be reformulated as:

[0089]

[0090] Among them, the Hessian matrix is the second-order partial derivative of the vector field f of the system i with respect to the state vector y of the system, and can be expanded as:

[0091]

[0092] For PEPS, usually, f i is only related to a small number of system states. Therefore, the system matrix is quite sparse. Without loss of generality, assume that f i is only related to the system state y ni = [y1, y2,..., y ni T where n i << n. Therefore, the valid elements in the Hessian matrix are only the upper left block matrix in Equation (23):

[0093]

[0094] Furthermore, generally, the second-order partial derivative of f i with respect to y satisfies the continuity condition, and the order of differentiation can be exchanged. Therefore this shows that the Hessian matrix is symmetric, so only the upper triangular matrix or the lower triangular matrix needs to be calculated. Generally speaking, for its computational complexity drops from the original n 2 to (1 + n i )n i / 2. Furthermore, considering all the second-order partial derivatives of f i in Equation (21), where i = 1, 2,..., n, its computational complexity drops from the original n 3 to Since n i << n, therefore, the computational cost and storage required to calculate using matrix sparsity techniques are quite low. Substituting Equation (21) into Equation (20), and then substituting Equation (20) into Equation (19), the calculation result of the eigenvalue sensitivity of the LTP system can be obtained.

[0095] Those skilled in the art can easily understand that the above is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.​

Claims

1. A method for fast calculation of sensitivity of linear periodic time-varying system based on matrix sparse technology, characterized by: include: Step S1, based on the Floquet-Lyapunov method, the LTP system is transformed into an equivalent LTI system through a periodic transformation matrix, and the stationary matrix of the LTI system is eigendecomposed to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system; Step S2, obtaining an analytical expression of the sensitivity of the LTP system eigenvalue to the parameters based on the functional derivative method; Step S3, distinguishing the direct and indirect effects of the parameters on the system matrix according to the mechanism of the effects of the parameters on the system matrix, and obtaining the analytical form of the LTP system matrix affected by the parameters in the sensitivity analytical expression; In step S4, the matrix sparsity technique is used to convert the partial derivative of the system matrix with respect to the steady-state trajectory into a sparse calculation of the vector field Hessian matrix, thereby obtaining the calculation result of the eigenvalue sensitivity of the LTP system.

2. The method for rapidly calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology according to claim 1, characterized in that: The sparse calculation of converting the partial derivative of the system matrix with respect to the steady-state trajectory into the vector field Hessian matrix in step S4 includes: Re-express the partial derivative of the system matrix with respect to the trajectory as And calculate the upper or lower triangular matrix of the Hessian matrix, where the Hessian matrix is the vector field f of the system i The second-order partial derivative with respect to the system's state vector y.

3. The method for fast calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology according to claim 2, characterized in that: In step S1, the LTP system is converted into an equivalent LTI system through a periodic transformation matrix, including: Get LTP System Where x(t)=[x1(t),x2(t),...,x n (t)] T represents the column vector composed of n state variables of the system at time t, A(t) represents the periodic system matrix, and T represents the basic period of the system; The periodic transformation matrix P(t) transforms the LTP system into an LTI system, and the systems before and after the transformation have the same stability: Among them, v(t) represents a set of column vectors consisting of new state variables, Q is a time-invariant n-order square matrix, and P(t) is an n-order square matrix with a period of T, thereby realizing the conversion of the time-invariant system matrix A(t) into a constant matrix Q.

4. The method for rapidly calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology according to claim 3 is characterized in that: In step S1, the LTI system's constant matrix is ​​subjected to eigendecomposition to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system, including: Perform eigendecomposition on the constant matrix Q: VQU=Λ=diag(λ1,λ2,...,λ n ), where V and U represent the left and right eigenvectors of the constant matrix Q, respectively, and the diagonal matrix Λ represents the Floquet characteristic index λ of the LTP system. i (i=1,2,...,n), i.e., LTP eigenvalue; Combining the transformation matrix P(t) with the right eigenvector U of the Q matrix, we can construct the coordinate transformation R(t)=P(t)U. The transformation matrix R(t) realizes that the system matrix A(t) remains unchanged and is converted into the eigenvalue matrix Λ of the LTP system: Where L(t) = R -1 (t) and R(t) represent the left and right eigenvectors of the LTP system, respectively.

5. The method for fast calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology according to claim 4 is characterized in that: The analytical expression for obtaining the sensitivity of the LTP system eigenvalue to the parameters in step S2 includes: In the LTP system, the mode λ k The first-order variation of dl k =λ k [α(t)+δα(t)]-λ k [α(t)], According to the functional derivative, the analytical expression of sensitivity is obtained as follows: When only time-invariant parameters are considered, In unit 1 form, the expression is 6. The method for fast calculating the sensitivity of a linear periodic time-varying system based on matrix sparse technology according to claim 5, characterized in that: The analytical form of the LTP system matrix obtained in step S3 includes: Denote the system matrix as A(y ss (α),α), where y ss is the steady-state solution; The influence of parameter α on the system matrix is ​​divided into indirect influence and direct influence, where the indirect influence A I The parameter affects the steady-state solution y ss (α) further affects the system matrix and directly affects A E The parameters directly affect the system matrix without affecting the steady-state solution; Based on the parameters of A I With A E The influence of the two parts, the analytical form of the parameter influence LTP system matrix is ​​obtained as follows:

7. A device for implementing the method for rapidly calculating the sensitivity of a linear periodic time-varying system based on matrix sparsity technology as described in any one of claims 1 to 6, characterized in that: include: The first processing module, based on the Floquet-Lyapunov method, transforms the LTP system into an equivalent LTI system through a periodic transformation matrix and performs eigendecomposition on the stationary matrix of the LTI system to obtain the eigenvalues ​​and left and right eigenvectors of the LTP system; The second processing module obtains the analytical expression of the sensitivity of the LTP system eigenvalue to the parameters based on the functional derivative method; The third processing module distinguishes the direct and indirect effects of parameters on the system matrix according to the mechanism of their influence on the system matrix, and obtains the analytical form of the parameter-affected LTP system matrix in the sensitivity analytical expression; The fourth processing module uses matrix sparsity technology to convert the partial derivatives of the system matrix with respect to the steady-state trajectory into a sparse calculation of the vector field Hessian matrix, thereby obtaining the calculation result of the eigenvalue sensitivity of the LTP system.

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